Merge these two meshes and export as .obj
When joining [&#9096; CtrlJ] objects, the new object will have the modifier stack of the active object.

If you (e.g.) select the mesh with the modifier and then the mesh without, the last selected becomes the active object (in this case the mesh without the modifier). When joining them, the new object will have no modifier and the mesh part which needs the modifier will be wrong.

The solution is to apply the modifier first by selecting the mesh and pressing apply on the modifier.



(legoparrot beat me, still going to leave the answer since there are some more details)